In this appendix we provide a glossary of the variables used in the text. For most terms we give a short explanation and refer to the equation or section where this variable is rst used. Usually it is dened there. If not, this should be a very common variable found, e.g., in most basic text books on general relativity (like the Christoel symbols, the Riemann tensor and so on).
A
Perturbation of the 00 component of the metric, respectively the lapse function (2.4),(2.11), Appendix A.B
Scalar perturbation of the 0i
component of the metric, respectively the shift vector (2.5),(2.11).B
i Vector perturbation of the 0i
component of the metric, respectively the shift vector (2.7),(2.12).B
ij Magnetic part of the Weyl tensor (2.27).C
=R
?(1=
2)(g
R
?+g
R
?g
R
?g
R
) +R6(g
g
?g
g
), Weyl tensor (2.26,27).D
() Gauge invariant density perturbation variable for the matter component (2.38).D
()g Gauge invariant density perturbation variable for the matter component (2.37).D
()s Gauge invariant density perturbation variable for the matter component (2.36).E
ij Electrical part of the Weyl tensor (2.26).F
Gauge dependent perturbation variable for the distribution function, paragraph (2.3.1).F(S) Gauge invariant perturbation variable for scalar perturbations of the distribution function (2.69).
F(T) Gauge invariant perturbation variable for tensor perturbations of the distribution function, paragraph (2.3.2).
F(V ) Gauge invariant perturbation variable for vector perturbations of the distribution function, paragraph (2.3.2).
G
Newtons constant,G
= 6:
672010?8cm
3g
?1sec
?2.G
=R
?(1=
2)g
R
, Einstein tensor.H
L Trace perturbation of the spatial part of the metric.H
T Anisotropic scalar perturbation of the spatial part of the metric (2.6,11).H
i Anisotropic vector perturbation of the spatial part of the metric (2.8,12).H
ij Anisotropic tensor perturbation of the spatial part of the metric (2.9,13).K
ij Second fundamental form, Section 2.1, Appendix A.L
i Spatial components of the vector eldX
parametrizing a gauge transformation, Section 2.1, 2.3.L
X Lie derivative w.r.t the vector eldX
, Section 2.3.M
Mass used to parametrize the energy momentum tensor of seed perturbations, Section 2.5.M Gauge invariant perturbation variable for the energy integrated photon distribution (2.83).
M Spacetime manifold, Appendix A, Section 2.3
M
i = 43 Rd
iM The rst moment ofM(3.19).M
ij = 83 Rd
ijMThe second moment of M(3.17).P
=u
u
+ The projection operator onto the 3{space orthogonal tou
(2.30).P
m The mass bundle, Section 2.3R
The Ricci scalar.R
= 3m=
4r Parameter used in paragraph 3.2.3.R Perturbation of the scalar curvature on the slices of constant time (2.14).
R
=R
The Ricci tensor.R
The Riemann tensorT
Temperature of the cosmic background radiation.T
Temporal component of the vector eldX
parametrizing a gauge transformation, Section 2.1, 2.3.T
M Tangent space to spacetime, Section 2.3.TX
Tangent vector eld associated to the vector eldX
, Section 2.3.1T
(sS) Scalar contribution to the energy momentum tensor of the seeds (2.117,118,119).T
(sT) Tensor contribution to the energy momentum tensor of the seeds (2.122).T
(sV ) Vector contribution to the energy momentum tensor of the seeds (2.120,121).V
Gauge invariant variable for scalar perturbations of the velocity eld (2.35).V
i Gauge invariant variable for vector perturbations of the velocity eld (2.42).X
Vector eld parametrizing a gauge transformation, Section 2.1, 2.3.@
= @x@ Partial derivative (vector eld)a
Cosmic scale factor, Section 1.1.b
Impact parameter, Section 3.4, 4.2.2.c
Speed of light, usually set equal to 1 in this text.c
s=pp=
_ _ , (c
=pp
_=
_) Adiabatic sound speed (of matter component), Section 1.1.e
Tetrad vector eld, Section 2.3, Appendix A.f
Distribution function of phase space, Section 2.3.f
Gauge invariant scalar potential parametrizing anisotropic stresses of seeds (2.119).f
Gauge invariant perturbation variable parametrizing the energy density of seeds (2.117).f
p Gauge invariant perturbation variable parametrizing the pressure of seeds (2.119).f
v Gauge invariant perturbation variable parametrizing the scalar velocity potential of seeds (2.118).g
Metric of spacetime, Chapter 2.h
Used to parametrize Hubble's constantH
0 =h
100secMpckm , Section 1.1.h
Metric perturbation (2.10).h
Planck's constant,h
= 1:
054610?27cm
2gsec
?1, usually set equal to 1 in this text.k
Spatial curvature of a Friedmann universe (1.1).k
Comoving wave number, Section 3.2, paragraph 3.4.2.k
B Boltzmann's constant,k
B= 1:
380710?16erg=K
, usually set equal to 1 in this text.l
Length introduced to keep perturbation variables dimensionless, in applications it may be set equal to a typical scale of perturbations, Section 2.1.l
H =t
Comoving size of the horizon, Section 1.3.q
Redshift corrected energy, paragraph 2.3.1.t
Conformal time, Section 1.1.t
T Conformal Thomson mean free path, Section 3.2.u
Energy velocity eld, Section 2.1v
Scalar velocity potential Section 2.1.v
Redshift corrected momentum, paragraph 2.3.1.v
i Vector peculiar velocity eld, Section 2.1.w
Enthalpy, Section 1.1.w
i() Gauge invariant vector potential parametrizing anisotropic stresses of seeds (2.121).w
i(v) Gauge invariant vector contribution to the energy ow of seeds (2.120).z
Cosmological redshift.? Gauge invariant entropy perturbation variable, Section 2.1.
? Christoel symbols, Section 2.3.
Laplacian.
Cosmological constant (1.1).
Gauge invariant scalar potential for anisotropic stresses, Section 2.1.
i Gauge invariant vector potential for anisotropic stresses, Section 2.1.
ij Gauge invariant tensor contribution to anisotropic stresses, Section 2.1.
Three dimensional spatial hypersurface, Appendix A.
Gauge invariant scalar potential for geometry perturbations (2.24).
Gauge invariant scalar potential for geometry perturbations (2.25).
i Gauge invariant perturbation variable for the uid vorticity (2.43).
Curvature 2{form, Appendix A.
Lapse function (2.4), Appendix A. Shift vector (2.5), Appendix A. ij Metric of a three space of constant curvature, Section 1.1. Gauge dependent density perturbation (2.28).i Spatial unit vector (e.g. denoting photon directions), Section 2.3.
= 4GM
2 Smallness parameter for the amplitude of seed perturbations, paragraph 2.5.2. ijk Three dimensional totally antisymmetric tensor (2.27). ij=ij?ij , Section 3.2. Symmetry breaking scale (4.1). Orthonormal tetrad of 1{forms, Appendix A.#
i Orthonormal triad of 1{forms on the hypersurfaces of constant time, Appendix A. Isomorphism between the perturbed and unperturbed mass bundles, Section 2.3. Gauge dependent perturbation variable for the energy integrated photon distribution para-graph (2.3.4). Parameter in the scalar eld potential (4.1). = (1 +kr
2=
4)?1 Conformal factor for the metric of a 3 space of constant curvature, Section 2.3. Cosine between the photon direction and the radial direction, paragraph 3.2.1. Orthonormal momentum components Section 2.3. ij Anisotropic stresses (2.31). L Gauge dependent pressure perturbation variable (2.31). (), () Background energy density of component . Scalar potential for the shear of the equal time hypersurfaces, extrinsic curvature (2.15). i Vector potential for the shear of the equal time hypersurfaces, extrinsic curvature (2.17). T Thomson cross section,T = 6:
652410?25cm
2. Physical time, Section 1.1. Impact time (4.30) Optical depth, Section 5.1. Stress tensor (2.30). Scalar eld, Chapter 4. Variable parametrizing spherically symmetric scalar eld congurations (4.4).!
Winding number density of the scalar eld, Section 4.1.!
Connection forms, Appendix 1.!
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Fig. 1:
The spectrum of the cosmic microwave background radiation as measures by COBE (Figure from Mather et al. [1990]).Fig. 2:
Limits on the CMB anisotropy on dierent angular scales. The COBE result is the only positive detection. All other marks represent 95% condence upper limits.Fig. 3:
The linear perturbation spectrum of hot dark matter, for one (2) or three (1) types of massive neutrinos. The uctuations are heavily damped on scales smaller than FSm
Pl=m
2 (Figure from Durrer [1989]).Fig. 4:
Simulations of structure formation with HDM (top right and bottom pictures) compared with the corresponding picture from the CfA survey (top left). Triangles are high density regions identied as galaxies. One sees that the simulations lead to highly over developed large scale structure (Figure from White [1986]).Fig. 5:
CDM simulations (a) and (b) compared with the CfA survey (c). No striking inconsistencies are visible at rst sight (Figure from Kolb and Turner [1990]).Fig. 6:
The angular galaxy galaxy correlation function as measured by the IRAS survey (black dots) compared with the predictions from CDM models withh
= 0:
4 (black line) andh
= 0:
5 (dotted line). The open circles and squares are results from an older analysis of the Lick catalogue (Figure from Maddox et al. [1990]).Fig. 7
The CMB anisotropy (in units of 10?3) from a spherically symmetric texture collapsing atz
= 30 (left) andz
= 200 (right) respectively as a function of angular separation from the center of the texture. This gure is calculated for a universe which reionizes atz
= 200. It shows how signals from small scale textures are substantially damped and broadened by photon diusion.Fig. 8:
The hot spot|cold spot signal of a spherically symmetric collapsing texture in units of 2:
810?4 . The horizontal variable =t
?r
cosdenotes the 'impact time' of a photon arriving at a distancer
from the texture at timet
traveling with an anglewith respect to the radial direction.The hot spot{cold spot is shown for photons with xed impact parameter
b
=r
sin0:
1t
c(t
cis the time of texture collapse). The signal from the expanding universe att
=t
c, line (1), andt
= 1:
5t
c, line (2), is compared with the at space result (dashed curve). The second peak appearing att
= 1:
5t
cis due to the dark matter potential.